Assertion -Let A - 0 c -b- -c 0 a- b -a 0 - and t- C- adjtI-A -t2I -tA -A2 -a2-b2- c2I Reason- adjA - A2 -a2- b2- c2I

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Special Integrals - 1

Assertion :Le...

Question

Assertion :Let $A = ⎡ ⎢ ⎣ \begin{matrix} 0 & c & - b - c & 0 & a b & - a & 0 \end{matrix} ⎤ ⎥ ⎦$ and $t \in C$ .
$a d j (t I - A) = t^{2} I + t A + A^{2} + (a^{2} + b^{2} + c^{2}) I$
Reason: $a d j A = A^{2} + (a^{2} + b^{2} + c^{2}) I$

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
$A = ⎡ ⎢ ⎣ \begin{matrix} 0 & c & - b - c & 0 & a b & - a & 0 \end{matrix} ⎤ ⎥ ⎦$
$\Rightarrow (t I - A) = ⎡ ⎢ ⎣ \begin{matrix} t & - c & b c & t & - a - b & a & t \end{matrix} ⎤ ⎥ ⎦$
$a d j (t I - A) = ⎡ ⎢ ⎣ \begin{matrix} t^{2} + a^{2} & t c + a b & a c - t b t c - a b & t^{2} + b^{2} & a t + b c a c + t b & - a t + b c & t^{2} + c^{2} \end{matrix} ⎤ ⎥ ⎦$
$= t^{2} ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦ + t ⎡ ⎢ ⎣ \begin{matrix} 0 & c & - b - c & 0 & a b & - a & 0 \end{matrix} ⎤ ⎥ ⎦ + ⎡ ⎢ ⎣ \begin{matrix} - c^{2} - b^{2} & a b & a c a b & - c^{2} - a^{2} & b c a c & b c & - b^{2} - a^{2} \end{matrix} ⎤ ⎥ ⎦ + (a^{2} + b^{2} + c^{2}) ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ a d j (t I - A) = t^{2} I + t A + A^{2} + (a^{2} + b^{2} + c^{2}) I$
Assertion is true.
$a d j A = ⎡ ⎢ ⎣ \begin{matrix} a^{2} & a b & a c a b & b^{2} & b c a c & b c & c^{2} \end{matrix} ⎤ ⎥ ⎦$
$= ⎡ ⎢ ⎣ \begin{matrix} - c^{2} - b^{2} & a b & a c a b & - c^{2} - a^{2} & b c a c & b c & - b^{2} - a^{2} \end{matrix} ⎤ ⎥ ⎦ + (a^{2} + b^{2} + c^{2}) ⎡ ⎢ ⎣ \begin{matrix} 1 & 0 & 0 0 & 1 & 0 0 & 0 & 1 \end{matrix} ⎤ ⎥ ⎦$
$∴ a d j A = A^{2} + (a^{2} + b^{2} + c^{2}) I$
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion.

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Similar questions

Q. Assertion: The compressibility factor for

H_{2}

and

H e

(1 + \frac{P b}{R T})

Reason: The compressibility factor for

H_{2}

and

H e

can be derived from Van der Waal's equation.

Assertion (A) Order of the reaction can be zero or fractional.

Reason (R) We cannot determine order from balanced chemical equation.

(a) Both assertion and reason are correct and the reason is correct explanation of assertion
(b) Both assertion and reason are correct, but reason does not explain assertion.
(c) Assertion is correct, but reason is incorrect
(d) Both assertion and reason are incorrect
(e) Assertion is incorrect but reason is correct.

Assertion (A) Order and molecularity are same.

Reason (R) Order is determined experimentally and molecularity is the sum of the stoichimetric coeffcient of rate determining elementary step.

Assertion (A) Boiling points of alcohols and ethers are high.
Reason (R) They can form intermolecular hydrogen-bonding.
(a) Assertion and reason both are correct and reason is correct explanation of assertion.
(b) Assertion and reason both are wrong statements.
(c) Assertion is correct statement but reason is wrong statement.
(d) Assertion is wrong statement but reason is correct statement.
(e) Both assertion and reason are correct statements but reason is not correct explanation of assertion.

ASSERTION: n-Pentane and n-Hexane belong to a homologous series.

REASON: n-Pentane and n-Hexane belong to the same class of hydrocarbons and they differ by a - $C H_{2}$ - group.

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Assertion :Let A=⎡⎢⎣0c−b−c0ab−a0⎤⎥⎦ and t∈C.adj(tI−A)=t2I+tA+A2+(a2+b2+c2)I Reason: adjA=A2+(a2+b2+c2)I

Assertion :Let $A = ⎡ ⎢ ⎣ \begin{matrix} 0 & c & - b - c & 0 & a b & - a & 0 \end{matrix} ⎤ ⎥ ⎦$ and $t \in C$ .
$a d j (t I - A) = t^{2} I + t A + A^{2} + (a^{2} + b^{2} + c^{2}) I$
Reason: $a d j A = A^{2} + (a^{2} + b^{2} + c^{2}) I$